This thesis investigates numerical methods for pricing options and forward contracts in Continuous Autoregressive Moving Average (CARMA) models. Via the Feynman-Kac connection between Markovian processes and Partial Differential Equations (PDEs), option and forward prices are obtained as solutions to their respective backward pricing PDEs. Our objective is to devise efficient numerical procedures that accommodate the possibly multi-dimensional CARMA state dynamics. We consider finite difference schemes and propose an Alternating Direction Implicit (ADI) method to deal with two or more spatial dimensions. The schemes are tested in a series of applications to the electricity and temperature markets for various spot models driven by CARMA processes. We discuss specifically CAR(1) and CARMA(2,1) processes, and illustrate how to adapt the finite difference schemes to solve so-called Partial Integro Differential Equations (PIDEs) to obtain forward prices in a jump-diffusion spot electricity model. Analytical results and Monte Carlo are used to benchmark the numerical approximations.